Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 79 / 4 / a / i

Use the method of multiple scales, $T=\epsilon t$, and write $x_0=A(T)\cos \psi$, $\psi =t+\theta (T)$. At first order,Removing the fundamental secular terms gives the averaged amplitude for position-dependent damping:where the brackets mean a full-period average. The initial conditions set $A(0)=1$ and $\theta (0)=0$ at leading order. For quadratic damping, $⟨{\sin }^2\psi ⟩=1/2$ and $⟨{\cos }^2\psi {\sin }^2\psi ⟩=1/8$, givingThus the Van der Pol amplitude evolution yieldsThe leading amplitude approaches the stable value two. The small initial-velocity discrepancy introduced by the slow amplitude derivative is removed by the first-order correction and does not affect this leading approximation.